In the 1920's, screw threads were standardized just as they are today. The one you just unscrewed is a #8-36.

All the screws are machine screws, using fine threads, and are slotted. Phillips head screws had not yet been invented!

The heads of the 1920's screws are basically cylinders. Today, the equivalent would be called a pan-head screw.

I'm not sure the lengths were standardized. If they were, I'm not too impressed with their accuracy. This one is 0.298 inches long (the threaded length), which is 19/64 inches. Yeah, they don't make screws in those sizes. If you want a replacement, go with the closest length.